1. Scaling Behavior in the Stochastic 1D Map
Eui-Sun Lee
Department of Physics
Kangwon National University
Stochastic 1D map
x : state variable
A : parameter controlling the degree of nonlinearity
 : parameter controlling the amplitude of the additive noise
Random numbers :
 : random variable,
n is statistically independent bounded random number,
for any n,
for nm.

2. Effect of Noise for period doubling Transition to Chaos
Bifurcation diagram & Lyapunov exponent
n is uniformly distributed over [-0.5,0.5], so
Bifurcation diagram
Lyapunov exponent
- <  < : Gray,
 > 0(Chaos) : Black.

3. Scaling factor for the stochastic 1D map
Parameter scaling factor: =
Orbital scaling factor:
= … .
Amplitude of noise scaling factor:
= … .
Through the renormalization group analysis , the amplitude of noise scaling factor is obtained.

4. Summary = 6.62903… . Amplitude of noise scaling factor:
1. Through the renormalization group analysis , The Amplitude of noise scaling factor(µ) is obtained.
Amplitude of noise scaling factor:
= … .